Opinions of Doron Zeilberger

53 点作者 kaladin-jasnah超过 1 年前

15 条评论

Animats超过 1 年前

His argument against infinities is interesting. He's making an argument for finite constructive mathematics. It's interesting to see that coming around again.Several decades ago, when I was working on early program proof of correctness, I spent a lot of time with the original Boyer-Moore theorem prover. This is a constructive theory of mathematics, similar to the Piano axioms. It really is possible to have useful mathematics without undecidablity. But you have to give up infinity. Boyer-Moore theory has recursion, but all recursions must have an upper bound. There must be some nonnegative integer that gets smaller with each recursion.Set theory is possible in the Boyer-Moore system. Boyer and Moore usually added the traditional axioms of set theory. But I tried doing without that to get to a McCarthy-like theory of arrays without set theory first. Arrays had a concrete representation as an ordered sequence of (index, value) tuples. The sequence has to be ordered so that array equality works. So all the functions for operating on arrays are rather clunky. Here's the sequence of machine proofs for that.[3]Back then, mathematicians hated such approaches. Infinity is a useful trick for handling the edge cases within the main case. Formulas become simpler. You can do things on blackboards with chalk more easily. Constructive math is boring and kind of ugly, because there's a lot of case analysis. But it's sound. You can avoid undecidability and Russel's class of all classes paradox. You don't actually need infinities for much of mathematics.Today there's more acceptance of machine proofs and case analysis. Everybody has computers, after all. I'm not a mathematician; I just used formal math as a tool to improve programs. This is a problem for the theorists.(A few years ago, I converted the original Boyer-Moore code to run under Gnu Common LISP, so it can be run today.[2] Proving the basics of number theory now takes under a second. It took about 45 minutes in 1981.)[1] <a href="https://sites.math.rutgers.edu/~zeilberg/Opinion160.html" rel="nofollow">https://sites.math.rutgers.edu/~zeilberg/Opinion160.html</a>[2] <a href="https://github.com/John-Nagle/nqthm">https://github.com/John-Nagle/nqthm</a>[3] <a href="https://github.com/John-Nagle/pasv/blob/master/src/work/temporaryrulebaseprooflog.txt">https://github.com/John-Nagle/pasv/blob/master/src/work/temp...</a>

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gurchik超过 1 年前

I might not have finished my Computer Science degree a few years ago if it wasn't due to Dr Z (as we called him at the time).I was required to take Linear Algebra to satisfy the requirements for the B.S. in Computer Science, but I simply could not get interested in the course. I had attempted to take this course 3 times previously, but could not succeed. Twice, I withdrew from the course before the first exam. The third time, I missed that window and ended up failing the class. I just couldn't be motivated to study this boring subject. A flaw of mine, as I've discovered. But in my 4th attempt, it was taught by Dr. Z, and it was completely different. His passion for mathematics (and more importantly, teaching) was infectious. This was completely different than my experience with some other professors in the Math and CS departments. I never got a feeling that Dr. Z was "wasting" his time on this 200-level course when he could be doing much more important research. I easily got an A, and with that, satisfied all the math and CS requirements for my degree. Years later, I remember a lot more about that class than, say, Calculus.

nickdrozd超过 1 年前

Zeilberger is a hardline finitist. According to him, a lot of mathematics is meaningless or at least woefully misguided due to its reliance on infinite sets, which he believes to be nonsense. His opinions are far from mainstream, but they sure are entertaining. Here's a taste:> But just like Gödel, [Turing] missed the point! Neither of them proved that "there exist true yet unprovable statements". Rubbish! Every meaningful statement is either provable (if it is true) or disprovable (if it is false). What they did (meta-)prove, by a Reductio Ad Absurdum clever argument, is that many statements that were believed to be meaningful, are really utterly devoid of meaning. As I have already preached in this sermon, the problem is the fictional (and pernicious!) infinity. In particular, Turing thesis's is utter nonsense, talking about "oracles", that lead to lots of beautiful, but fictional and irrelevant work by logicians and theoretical computer scientists.

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pessimizer超过 1 年前

<a href="https://sites.math.rutgers.edu/~zeilberg/Opinion8.html" rel="nofollow">https://sites.math.rutgers.edu/~zeilberg/Opinion8.html</a>"[...]I was shocked the other day, when I was browsing in the Web, that the AMS journals are not available freely for downloading, but charge subscription fees. Some `non-profit' organization! Let me remind you that there are several excellent electronic journals, that are viewable free of charge, for example New York Journal of Mathematics and the Electronic J. of Combinatorics. If you publish your papers there, you would be accessible to the ten millions people who surf the Web regularly. Of course, if you are already a tenured full professor, it is even pointless to publish there, since a publication reachable from your Home Page is equally accessible, although the journals might reach more browsers."Let's hope that the new medium will destroy the old organizations with their corrupt paper mentality and elitism, or at least reform them. After all, thanks to the Web and E-mail, both journals and conferences are becoming increasingly pointless. Then again, institutions have a very strong inertia, look at established religion, so perhaps, unfortunately, it is too soon to rejoice in the devil's death."Opinion 8: Organized Mathematics = Organized Crime, Dec. 1, 1995

paulpauper超过 1 年前

"Opinion 185: David Jackson and Bruce Richmond Should Retract Their Erroneous Attempted Proof of the Four-Color-Theorem [Written Dec. 30, 2022]"Of course the arxiv is full of claimed proofs of Collatz and other major open problems, but there is a special section for them called GM (General Mathematics).It really damages the credibility of the arxiv if a non-GM department, in this case CO (combinatorics) has a false proof of a long-standing open problem. I am talking about the recent Erroneous computer-less proof of the Four Color Theorem posted by distinguished and accomplished combinatorialists David Jackson and Bruce Richmond, who hail from the Mecca of combinatorics, University of Waterloo. Their proof is definitely flawed in its current form, and very unlikely to be fixable. (See here).And before that:"appendix:Why ArXiv Moderators (including Victor Reiner) Seriously Erred in Rejecting a One Page Gem Entitled "Five More Proofs of the Cosine Addition Formula (Inspired by Mark Levi's Perpetuum Mobile Proof)"He wanted arxiv to reject or reclassify the flawed four-color proof, but mad that arxiv rejected his paper. In the first instance, the pre-print processes worked as it was intended to. The paper was uploaded , critiqued, and then retracted. it would seem like he wants to be the arbiter of which papers get rejected/reclassified on arXiv.

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jasperry超过 1 年前

It's very interesting that he is strongly for the use of computation in math, but strongly against using computer logic systems to formalize proofs--see opinion #184. I'd love to hear other people's take on this.I saw how pro-computational he is from the one time I heard him talk. One of his most repeated phrases was, "you write a little computer program...", meaning as a way to enumerate some set of objects or calculate the size of something.

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Tainnor超过 1 年前

How long would the following program run?<pre><code> while (true) {} </code></pre> I assume an ultrafinitist would argue that the question is ill-posed. It will always depend on the exact hardware that the program runs on (as every hardware will eventually fail). Or on us knowing when the heat death of the universe will be (which gives an upper bound).But that means that suddenly a theory of computability must deal with physical concerns and can't even talk about an abstraction such as "program" anymore. This seems completely contrary to how even just practically minded programmers who aren't deep into infinitary maths would think about their activity.Yes, you can make a philosophical argument for ultrafinitism - I would probably also say that arbitrarily large numbers don't "actually" exist in reality - but you just throw away so much useful abstraction that I honestly don't understand what the point is.

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dang超过 1 年前

Related:The Mathematical Opinions of Dr. Doron Zeilberger - <a href="https://news.ycombinator.com/item?id=485899">https://news.ycombinator.com/item?id=485899</a> - Feb 2009 (8 comments)

BeetleB超过 1 年前

Doron's opinions are always fun to read.People refer to him as the official mathematics troll (in a nice way).

zero-sharp超过 1 年前

I'm pretty sure that making arguments under idealistic assumption often allows us to make progress. Yea certain methods can seem problematic and I definitely have worried about this before. But I'm pretty sure there's evidence that certain, useful, theories just wouldn't exist unless we used infinity in some capacity.

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adrian_b超过 1 年前

<a href="https://sites.math.rutgers.edu/~zeilberg/Opinion158.html" rel="nofollow">https://sites.math.rutgers.edu/~zeilberg/Opinion158.html</a>" So brace yourself. In two years, you would open-up an AMS journal, and read an abstract that looks like this:<pre><code> "Using Theory T1014, and Theorems Th100154, Th54135, and Th87651, as well as inequalities In54312, and In34567, we prove Conjecture C84231." </code></pre> Please! Mathematics papers are hard enough to read the way they are written now, and this new policy would make reading them so much harder. It is true that few people read papers any more, and eventually, only computers will write and read papers, and for them using numbered-theorems would be preferred, so let's postpone this reform for another fifty years, when humans will no longer care how unreadable a mathematics paper is"

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bigbillheck超过 1 年前

I don't usually agree with Zeilberger, but his heart's in the right place and I do, in fact, have to hand it to him.

Vecr超过 1 年前

See Gödel, Escher, Bach [1979, pp. 452–456]) (that's pages 452 to 456 in the original) for more about problems caused by infinite precision, infinite numbers, and infinite "tape size" (for Turing machines). I and other people would also add "infinite time" as well.

yding超过 1 年前

Great to see. Not only a wonderful mathematician but also a wonderful human being.

ykonstant超过 1 年前

I love reading Doron's opinions, whether I agree with them or not.